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Pietro Miraglio

Publications and source records attributed to Pietro Miraglio.

10 recordsLinked to original sources
arXiv2025

Deep Learning for Sea Surface Temperature Reconstruction under Cloud Occlusion

Sea Surface Temperature (SST) reconstructions from satellite images affected by cloud gaps have been extensively documented in the past three decades. Here we describe several Machine Learning models to fill the cloud-occluded areas starting from MODIS Aqua nighttime L3 images. To tackle this challenge, we employed a type of Convolutional Neural Network model (U-net) to reconstruct cloud-covered portions of satellite imagery while preserving the integrity of observed values in cloud-free areas. We demonstrate the outstanding precision of U-net with respect to available products done using OI interpolation algorithms. Our best-performing architecture show 50% lower root mean square errors over established gap-filling methods.

cs.CV↗
arXiv2022

Optimal regularity of stable solutions to nonlinear equations involving the $p$-Laplacian

We consider the equation $-Δ_p u=f(u)$ in a smooth bounded domain of $\mathbb{R}^n $, where $Δ_p$ is the $p$-Laplace operator. Explicit examples of unbounded stable energy solutions are known if $n\geq p+4p/(p-1)$. Instead, when $n<p+4p/(p-1)$, stable solutions have been proved to be bounded only in the radial case or under strong assumptions on $f$. In this article we solve a long-standing open problem: we prove an interior $C^α$ bound for stable solutions which holds for every nonnegative $f\in C^1$ whenever $p\geq2$ and the optimal condition $n<p+4p/(p-1)$ holds. When $p\in(1,2)$, we obtain the same result under the non-sharp assumption $n<5p$. These interior estimates lead to the boundedness of stable and extremal solutions to the associated Dirichlet problem when the domain is strictly convex. Our work extends to the $p$-Laplacian some of the recent results of Figalli, Ros-Oton, Serra, and the first author for the classical Laplacian, which have established the regularity of stable solutions when $p=2$ in the optimal range $n<10$.

math.AP↗
arXiv2022

A data filling methodology for time series based on CNN and (Bi)LSTM neural networks

In the process of collecting data from sensors, several circumstances can affect their continuity and validity, resulting in alterations of the data or loss of information. Although classical methods of statistics, such as interpolation-like techniques, can be used to approximate the missing data in a time series, the recent developments in Deep Learning (DL) have given impetus to innovative and much more accurate forecasting techniques. In the present paper, we develop two DL models aimed at filling data gaps, for the specific case of internal temperature time series obtained from monitored apartments located in Bolzano, Italy. The DL models developed in the present work are based on the combination of Convolutional Neural Networks (CNNs), Long Short-Term Memory Neural Networks (LSTMs), and Bidirectional LSTMs (BiLSTMs). Two key features of our models are the use of both pre- and post-gap data, and the exploitation of a correlated time series (the external temperature) in order to predict the target one (the internal temperature). Our approach manages to capture the fluctuating nature of the data and shows good accuracy in reconstructing the target time series. In addition, our models significantly improve the already good results from another DL architecture that is used as a baseline for the present work.

cs.LG↗
arXiv2020

Universal Hardy-Sobolev inequalities on hypersurfaces of Euclidean space

In this paper we study Hardy-Sobolev inequalities on hypersurfaces of $\mathbb{R}^{n+1}$, all of them involving a mean curvature term and having universal constants independent of the hypersurface. We first consider the celebrated Sobolev inequality of Michael-Simon and Allard, in our codimension one framework. Using their ideas, but simplifying their presentations, we give a quick and easy-to-read proof of the inequality. Next, we establish two new Hardy inequalities on hypersurfaces. One of them originates from an application to the regularity theory of stable solutions to semilinear elliptic equations. The other one, which we prove by exploiting a "ground state" substitution, improves the Hardy inequality of Carron. With this same method, we also obtain an improved Hardy or Hardy-Poincaré inequality.

math.AP↗
arXiv2020

(Non)local $Γ$-convergence

We present some long-range interaction models for phase coexistence which have recently appeared in the literature, recalling also their relation to classical interface and capillarity problems. In this note, the main focus will be on the $Γ$-convergence methods, emphasizing similarities and differences between the classical theory and the new trends of investigation. In doing so, we also obtain some new, more precise $Γ$-convergence results in terms of "interior" and "exterior" contributions. We also discuss the structural differences between $Γ$-limits and "pointwise" limits, especially concerning the "boundary terms".

math.AP↗
arXiv2019

Energy asymptotics of a Dirichlet to Neumann problem related to water waves

We consider a Dirichlet to Neumann operator $\mathcal{L}_a$ arising in a model for water waves, with a nonlocal parameter $a\in(-1,1)$. We deduce the expression of the operator in terms of the Fourier transform, highlighting a local behavior for small frequencies and a nonlocal behavior for large frequencies. We further investigate the $ Γ$-convergence of the energy associated to the equation $ \mathcal{L}_a(u)=W'(u) $, where $W$ is a double-well potential. When $a\in(-1,0]$ the energy $Γ$-converges to the classical perimeter, while for $a\in(0,1)$ the $Γ$-limit is a new nonlocal operator, that in dimension $n=1$ interpolates the classical and the nonlocal perimeter.

math.AP↗
arXiv2019

Symmetry results for the solutions of a partial differential equation arising in water waves

This paper recalls some classical motivations in fluid dynamics leading to a partial differential equation which is prescribed on a domain whose boundary possesses two connected components, one endowed with a Dirichlet datum, and the other endowed with a Neumann datum. The problem can also be reformulated as a nonlocal problem on the component endowed with the Dirichlet datum. A series of recent symmetry results are presented and compared with the existing literature.

math.AP↗
arXiv2017

One-dimensional symmetry for the solutions of a three-dimensional water wave problem

We prove a one-dimensional symmetry result for a weighted Dirichlet-to-Neumann problem arising in a model for water waves in dimension 3. More precisely we prove that minimizers and bounded monotone solutions depend on only one Euclidean variable. The analogue of this result for the 2-dimensional case (and without weights) was established in an article by De La Llave and the third author. In this paper, a crucial ingredient in the proof is given by an energy estimate for minimizers obtained via a comparison argument.

math.AP↗
arXiv2016

The Phillip Island penguin parade (a mathematical treatment)

Penguins are flightless, so they are forced to walk while on land. In particular, they show rather specific behaviors in their homecoming, which are interesting to observe and to describe analytically. In this paper, we present a simple mathematical formulation to describe the little penguins parade in Phillip Island. We observed that penguins have the tendency to waddle back and forth on the shore to create a sufficiently large group and then walk home compactly together. The mathematical framework that we introduce describes this phenomenon, by taking into account "natural parameters" such as the eye-sight of the penguins, their cruising speed and the possible "fear" of animals. On the one hand, this favors the formation of conglomerates of penguins that gather together, but, on the other hand, this may lead to the "panic" of isolated and exposed individuals. The model that we propose is based on a set of ordinary differential equations. Due to the discontinuous behavior of the speed of the penguins, the mathematical treatment (to get existence and uniqueness of the solution) is based on a "stop-and-go" procedure. We use this setting to provide rigorous examples in which at least some penguins manage to safely return home (there are also cases in which some penguins freeze due to panic). To facilitate the intuition of the model, we also present some simple numerical simulations that can be compared with the actual movement of the penguins parade.

math.CA↗